Abstract and Applied Analysis

Volume 2014 (2014), Article ID 916260, 6 pages

http://dx.doi.org/10.1155/2014/916260

Research Article

Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions

Guotao Wang,¹ Sanyang Liu,¹ and Lihong Zhang²

¹Department of Applied Mathematics, Xidian University, Xi'an, Shaanxi 710071, China

²School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 26 November 2013; Accepted 13 February 2014; Published 24 March 2014

Academic Editor: Hossein Jafari

Copyright © 2014 Guotao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.

1. Introduction

Fractional calculus has been receiving more and more attention in view of its extensive applications in the mathematical modelling coming from physical and other applied sciences; see books [1–5]. Recently, the existence of solutions (or positive solutions) of nonlinear fractional differential equation has been investigated in many papers (see [6–28] and references cited therein). However, in terms of the eigenvalue problem of fractional differential equation, there are only a few results [29–33].

To the best of author’s knowledge, no paper has considered the eigenvalue problem of the following nonlinear fractional differential equation with integral boundary conditions:

where , is the Caputo fractional derivative, and is a continuous function.

Our proof is based upon the properties of the Green function and Guo-Krasnoselskii’s fixed point theorem given in [34]. Our purpose here is to give the eigenvalue interval for nonlinear fractional differential equation with integral boundary conditions. Moreover, according to the range of the eigenvalue , we establish some sufficient conditions for the existence and nonexistence of at least one positive solution of the problem (1).

2. Preliminaries

For the convenience of the readers, we first present some background materials.

Definition 1. For a function , the Caputo derivative of fractional order is defined as

where denotes the integer part of the real number .

Definition 2. The Riemann-Liouville fractional integral of order for a function is defined as

provided that such integral exists.

Lemma 3. Let ; then

for some ,  , .

Lemma 4 (see [34]). Let be a Banach space, and let be a cone. Assume that , are open subsets of with , , and let be a completely continuous operator such that

(i), , and , , or(ii), , and , .

Then has a fixed point in .

Lemma 5. Let , , , and . Assume ; then the unique solution of the problem

is given by the expression

where

Proof. It is well known that the equation can be reduced to an equivalent integral equation:

for some   .

By the conditions and , we can get that and

Hence, we have

Put ; then, from (10), we deduce that

which implies that

Replacing this value in (10), we obtain the following expression for function :

This completes the proof.

Lemma 6. Let be the Green function, which is given by the expression (7). For , the following property holds:

The proof is similar to that of Lemma 2.4 in [7], so we omit it.

Consider the Banach space with general norm

Define the cone .

Suppose is a solution of (1). It is clear from Lemma 5 that

Define the operator as follows:

Lemma 7. is completely continuous.

Proof. Since , it is obvious that . So we have

Therefore, . The other proof is similar to that in [7], so we omit it.

3. Main Result

For convenience, we list the denotation:

Next, we will establish some sufficient conditions for the existence and nonexistence of positive solution for problem (1).

Theorem 8. Let be a constant. Then for each

problem (1) has at least one positive solution.

Proof. First, for any , from (20) we have

On the one hand, by the definition of , there exists such that, for any , we have

Choose . For , we have

On the other hand, by the definition of , there exists such that, for any , we have

Take . For , we have

According to (23),  (25), and Lemma 4, has at least one fixed point with , which is a positive solution of (1).

Remark 9. If and , then we can get

Theorem 8 implies that, for , problem (1) has at least one positive solution.

Theorem 10. Let be a constant. Then for each

problem (1) has at least one positive solution.

Proof. First, it follows from (27) that, for any ,

By the definition of , there exists such that, for any , we have

Choose . For , we have . Similar to the proof in Theorem 8, it holds from (28) and (29) that

Note . There exists , such that

We consider the problem on two cases. (I) Suppose is bounded. There exists , such that , . Choose . Let . For , we have

(II) Suppose is unbounded. There exists such that

Let . For , we have

Combining (I) and (II), take ; here, . Then for , we have

Hence, (30) and (42) together with Lemma 4 imply that has at least one fixed point with , which is a positive solution of (1).

Theorem 11. Assume and . Problem (1) has no positive solution provided

where is a constant defined in (38).

Proof. Since and , together with the definitions of and , there exist positive constants , , , and satisfying such that

Take

It follows that for any . Suppose that is a positive solution of (1). That is,

In sequence,

which is a contradiction. Hence, (1) has no positive solution.

Theorem 12. Assume and . Problem (1) has no positive solution provided

where is a constant defined in (43).

Proof. Since and , together with the definitions of and , there exist positive constants , , , and satisfying such that

Take

It follows that for any . Suppose that is a positive solution of (1). That is,

In sequence,

which is a contradiction. Hence, (1) has no positive solution.

Example 13. Consider the fractional differential equation

In this example, take

Obviously, we have

Since and , through a computation, we can get

Choose ; we have

Theorem 8 implies that, for , , the problem (46) has at least one positive solution.

Remark 14. In particular, if we take in Example 13, then and . Remark 9 implies that problem (46) has at least one positive solution for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the NNSF of China (no. 61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 2012021002-3).

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